Differential EquationsDifferential Equations

What a differential equation is, what counts as a solution, and how direction fields show solutions before you solve anything.

1. 1Differential Equations: Definitions
2. 2Integrals as Solutions
3. 3Direction Fields

Linear, separable, exact and Bernoulli equations, substitutions, where solutions live, modeling, equilibria, and Euler's method.

1. 4Linear Equations
2. 5Separable Equations
3. 6Exact Equations
4. 7Bernoulli Equations
5. 8Substitutions
6. 9Intervals of Validity
7. 10Modeling with First Order Equations
8. 11Equilibrium Solutions
9. 12Euler's Method

Constant-coefficient equations and their three cases, the Wronskian, undetermined coefficients, variation of parameters, and mechanical vibrations.

1. 13Second Order Linear Equations
2. 14Real, Distinct Roots
3. 15Complex Roots
4. 16Repeated Roots
5. 17Reduction of Order
6. 18Fundamental Sets and the Wronskian
7. 19Undetermined Coefficients
8. 20Variation of Parameters
9. 21Mechanical Vibrations

Turning differential equations into algebra: the transform, its table, inverses, step functions, impulses and convolution.

1. 22The Laplace Transform
2. 23Computing Laplace Transforms
3. 24Inverse Laplace Transforms
4. 25Step Functions
5. 26Solving IVPs with Laplace Transforms
6. 27IVPs with Step Functions
7. 28The Dirac Delta Function
8. 29Convolution Integrals

Several interacting quantities at once: matrix form, eigenvalues, phase portraits, and models from ecology and epidemiology.

1. 30Systems of Differential Equations
2. 31Eigenvalues and Eigenvectors
3. 32The Phase Plane
4. 33Real Eigenvalues
5. 34Complex Eigenvalues
6. 35Repeated Eigenvalues
7. 36Modeling with Systems

When coefficients aren't constant: power series solutions, recurrence relations, and Euler equations.

1. 37Series Solutions
2. 38Euler Equations

Everything from second order, extended: characteristic polynomials of any degree and undetermined coefficients.

1. 39Higher Order Linear Equations
2. 40Higher Order Nonhomogeneous Equations

Conditions at two ends instead of one, eigenvalue problems, and writing functions as sums of sines and cosines.

1. 41Boundary Value Problems
2. 42Eigenvalues and Eigenfunctions
3. 43Periodic and Orthogonal Functions
4. 44Fourier Sine Series
5. 45Fourier Cosine Series
6. 46Fourier Series